From http://en.wikipedia.org/wiki/Wigner%E2%80%93Weyl_transform: "It is now understood that Weyl quantization is not always well defined and sometimes gives unphysical answers."
I am looking for references and examples for the statement above.
in a paper of Buchholz and Grundling (arXiv:0705.1988) you can find a discussion of drawbacks of Weyl quantization. In particular, they mention a result of Fannes and Verbeure (1974), stating that for nontrivial potentials the time-evolution induced by the Hamiltonian is not a *-automorphism of the Weyl algebra.